Unit Circle

The unit circle is the circle of radius $1$ centered at the origin in the coordinate plane: $x^2 + y^2 = 1$.

Its power is that it extends trigonometry beyond right triangles. For any angle $\theta$ measured counterclockwise from the positive x-axis, the point on the unit circle at that angle is $(\cos\theta, \sin\theta)$.

That single definition gives:

• $\sin\theta$ and $\cos\theta$ for all real $\theta$ (not just $0° < \theta < 90°$),
• The periodicity $\sin(\theta + 2\pi) = \sin\theta$,
• The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ (it's literally the equation of the circle),
• The signs of $\sin$ and $\cos$ in each quadrant.

Memorising the first quadrant key angles ($0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$) and using symmetry covers the entire circle. The unit circle is the most useful single picture in all of trigonometry — well worth a dedicated study session.